scholarly journals Cubic function fields with prescribed ramification

2021 ◽  
pp. 1-35
Author(s):  
Valentijn Karemaker ◽  
Sophie Marques ◽  
Jeroen Sijsling
Keyword(s):  
Rational Function ◽  
Function Field ◽  
Function Fields ◽  
Hyperelliptic Curves ◽  
Genus Zero ◽  
Isomorphism Classes ◽  
Explicit Procedure ◽  
Cubic Function Fields ◽  
Cubic Function ◽  
Elliptic And Hyperelliptic Curves

This paper describes cubic function fields [Formula: see text] with prescribed ramification, where [Formula: see text] is a rational function field. We give general equations for such extensions, an explicit procedure to obtain a defining equation when the purely cubic closure [Formula: see text] of [Formula: see text] is of genus zero, and a description of the twists of [Formula: see text] up to isomorphism over [Formula: see text]. For cubic function fields of genus at most one, we also describe the twists and isomorphism classes obtained when one allows Möbius transformations on [Formula: see text]. The paper concludes by studying the more general case of covers of elliptic and hyperelliptic curves that are ramified above exactly one point.

scholarly journals Tabulation of cubic function fields via polynomial binary cubic forms

2012 ◽  
Vol 81 (280) ◽  
pp. 2335-2359 ◽  
Author(s):  
Pieter Rozenhart ◽  
Michael Jacobson ◽  
Renate Scheidler
Keyword(s):  
Function Fields ◽  
Cubic Forms ◽  
Cubic Function Fields ◽  
Cubic Function

Defining transcendentals in function fields

2002 ◽  
Vol 67 (3) ◽  
pp. 947-956 ◽  
Author(s):  
Jochen Koenigsmann
Keyword(s):  
Rational Function ◽  
Function Field ◽  
Function Fields ◽  
Algebraic Closure ◽  
Rational Function Field ◽  
First Order

AbstractGiven any field K, there is a function field F/K in one variable containing definable transcendental over K, i.e., elements in F / K first-order definable in the language of fields with parameters from K. Hence, the model-theoretic and the field-theoretic relative algebraic closure of K in F do not coincide. E.g., if K is finite, the model-theoretic algebraic closure of K in the rational function field K(t) is K(t).For the proof, diophantine ∅-definability ofK in F is established for any function field F/K in one variable, provided K is large, or K× /(K×)n is finite for some integer n > 1 coprime to char K.

An Explicit Treatment of Cubic Function Fields with Applications

2010 ◽  
Vol 62 (4) ◽  
pp. 787-807 ◽  
Author(s):  
E. Landquist ◽  
P. Rozenhart ◽  
R. Scheidler ◽  
J. Webster ◽  
Q. Wu
Keyword(s):  
Function Fields ◽  
Standard Form ◽  
Irreducible Polynomial ◽  
Zeta Functions ◽  
Cube Root ◽  
Integral Bases ◽  
Root Extraction ◽  
Necessary And Sufficient Criteria ◽  
Cubic Function Fields ◽  
Cubic Function

AbstractWe give an explicit treatment of cubic function fields of characteristic at least five. This includes an efficient technique for converting such a field into standard form, formulae for the field discriminant and the genus, simple necessary and sufficient criteria for non-singularity of the defining curve, and a characterization of all triangular integral bases. Our main result is a description of the signature of any rational place in a cubic extension that involves only the defining curve and the order of the base field. All these quantities only require simple polynomial arithmetic as well as a few square-free polynomial factorizations and, in some cases, square and cube root extraction modulo an irreducible polynomial. We also illustrate why and how signature computation plays an important role in computing the class number of the function field. This in turn has applications to the study of zeros of zeta functions of function fields.

scholarly journals Quadratic sequences of powers and Mohanty’s conjecture

2018 ◽  
Vol 14 (02) ◽  
pp. 479-507 ◽  
Author(s):  
Natalia Garcia-Fritz
Keyword(s):  
Elliptic Curves ◽  
Arithmetic Progression ◽  
General Type ◽  
Function Fields ◽  
Hyperelliptic Curves ◽  
Arithmetic Progressions ◽  
Surfaces Of General Type ◽  
Genus Zero ◽  
Curves Of Low Genus

We prove under the Bombieri–Lang conjecture for surfaces that there is an absolute bound on the length of sequences of integer squares with constant second differences, for sequences which are not formed by the squares of integers in arithmetic progression. This answers a question proposed in 2010 by Browkin and Brzezinski, and independently by Gonzalez-Jimenez and Xarles. We also show that under the Bombieri–Lang conjecture for surfaces, for every [Formula: see text] there is an absolute bound on the length of sequences formed by [Formula: see text]th powers with constant second differences. This gives a conditional result on one of Mohanty’s conjectures on arithmetic progressions in Mordell’s elliptic curves [Formula: see text]. Moreover, we obtain an unconditional result regarding infinite families of such arithmetic progressions. We also study the case of hyperelliptic curves of the form [Formula: see text]. These results are proved by unconditionally finding all curves of genus zero or one on certain surfaces of general type. Moreover, we prove the unconditional analogues of these arithmetic results for function fields by finding all the curves of low genus on these surfaces.

Sign in / Sign up

Export Citation Format

Share Document